Nonlinear Volterra equation

Summary

A nonlinear Volterra equation is an integral equation $\varphi(t) = f(t) + \int_a^t K(t, s, \varphi(s))\, ds$. The upper limit of integration is the variable itself, and the unknown may enter the integrand nonlinearly. It extends the linear Volterra integral equation and contains every initial value problem $y' = F(t, y)$, $y(a) = y_0$, in integral form. When $K$ is Lipschitz in its last variable, successive approximations give a unique solution on the whole interval. When the Lipschitz bound holds only on bounded sets, a solution may exist only on a short interval.